Which of the following statements is logically equivalent to: "If he studies, he will pass the course." (A) He passed the course; therefore, he studied.
(B) He did not study; therefore, he will not pass the course.
(C) He did not pass the course; therefore he did not study.
(D) He will pass the course only if he studies.
(E) None of the above.

We have three types of logical statements that are formed from a conditional statement. They are:

Let's say the conditional statement is: If A, then B.

One important thing to note is that while these are all variations of a conditional statement, the converse and inverse are not always necessarily true.

On the other hand, the contrapositive is always true. This can be proved mathematically, and I will prove it now in an example.

A standard example would be the following conditional statement:

We agree that the original statement is true. Now, let's form the converse, inverse, and contrapositive to this statement:

While this may be true, it is not guaranteed that it rained last night. The sidewalk could be wet for other reasons than rain.

This statement obviously seems flawed, as there was no rain to make the sidewalk wet. It must be wet for other reasons.

This statement is true and is logically equivalent to the original statement.

Now, let's apply our knowledge of logically equivalent statements to the given conditional statement.

If he studies, he will pass the course.

The sequence of this statement, not-B then not-A, is followed in answer choice C. Therefore, this is the correct answer choice.

Remember that if a question asks you to find the logically equivalent statement to a conditional, always look for the contrapositive.

Explanation:

In logic, contrapositive of an implication is always true:

“A implies B” is equivalent to “¬B implies ¬A”

Applying that concept in this case:

"If he studies, he will pass the course."

The contrapositive is:

If he does not pass the course, he has not studied.

Among the choices, the answer is (C) He did not pass the course; therefore he did not study.

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